About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 786548, 8 pages
http://dx.doi.org/10.1155/2013/786548
Research Article

Weak Solutions for a -Laplacian Antiperiodic Boundary Value Problem with Impulsive Effects

1Department of Mathematics, Qilu Normal University, Jinan, Shandong 250013, China
2School of Mathematics, Shandong University, Jinan, Shandong 250100, China
3Department of Mathematics, Hebei University of Engineering, Handan, Hebei 056038, China

Received 7 March 2013; Accepted 19 May 2013

Academic Editor: Beatrice Paternoster

Copyright © 2013 Keyu Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989. View at Zentralblatt MATH · View at MathSciNet
  3. M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, vol. 2 of Contemporary Mathematics and Its Applications, Hindawi Publishing Corporation, New York, NY, USA, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. J. Nieto and D. O'Regan, “Variational approach to impulsive differential equations,” Nonlinear Analysis. Real World Applications, vol. 10, no. 2, pp. 680–690, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Sun, H. Chen, J. J. Nieto, and M. Otero-Novoa, “The multiplicity of solutions for perturbed second-order Hamiltonian systems with impulsive effects,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 12, pp. 4575–4586, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. Y. Tian, W. Ge, and D. Yang, “Existence results for second-order system with impulse effects via variational methods,” Journal of Applied Mathematics and Computing, vol. 31, no. 1-2, pp. 255–265, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. Xiao, J. J. Nieto, and Z. Luo, “Multiplicity of solutions for nonlinear second order impulsive differential equations with linear derivative dependence via variational methods,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 1, pp. 426–432, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. Zhou and Y. Li, “Existence of solutions for a class of second-order Hamiltonian systems with impulsive effects,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1594–1603, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. P. Chen and X. H. Tang, “Existence of solutions for a class of p-Laplacian systems with impulsive effects,” Taiwanese Journal of Mathematics, vol. 16, no. 3, pp. 803–828, 2012. View at MathSciNet
  10. P. Chen and X. Tang, “Existence and multiplicity of solutions for second-order impulsive differential equations with Dirichlet problems,” Applied Mathematics and Computation, vol. 218, no. 24, pp. 11775–11789, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  11. L. Bai and B. Dai, “Three solutions for a p-Laplacian boundary value problem with impulsive effects,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 9895–9904, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  12. L. Bai and B. Dai, “Existence and multiplicity of solutions for an impulsive boundary value problem with a parameter via critical point theory,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1844–1855, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. L. Wang, W. Ge, and M. Pei, “Infinitely many solutions of a second-order p-Laplacian problem with impulsive condition,” Applications of Mathematics, vol. 55, no. 5, pp. 405–418, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  14. I. Bogun, “Existence of weak solutions for impulsive p-Laplacian problem with superlinear impulses,” Nonlinear Analysis. Real World Applications, vol. 13, no. 6, pp. 2701–2707, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  15. T. Bartsch, “Infinitely many solutions of a symmetric Dirichlet problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 20, no. 10, pp. 1205–1216, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. T. Bartsch and M. Willem, “On an elliptic equation with concave and convex nonlinearities,” Proceedings of the American Mathematical Society, vol. 123, no. 11, pp. 3555–3561, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. Chen and X. Tang, “Infinitely many solutions for a class of fractional boundary value problem,” http://www.emis.de/journals/BMMSS/pdf/acceptedpapers/2011-09-043_R1.pdf.
  18. L. Zhang, X. H. Tang, and J. Chen, “Infinitely many periodic solutions for some second-order differential systems with p(t)-Laplacian,” Boundary Value Problems, vol. 33, 15 pages, 2011. View at MathSciNet
  19. U. B. Severo, “Multiplicity of solutions for a class of quasilinear elliptic equations with concave and convex terms in ,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 5, pp. 1–16, 2008. View at Zentralblatt MATH · View at MathSciNet
  20. S. Yan and J. Yang, “Fountain theorem over cones and applications,” Acta Mathematica Scientia B, vol. 30, no. 6, pp. 1881–1888, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  21. D. Liu, “On a p-Kirchhoff equation via fountain theorem and dual fountain theorem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 1, pp. 302–308, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  22. P. S. Iliaş, “Existence and multiplicity of solutions of a p(x)-Laplacian equation in a bounded domain,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 52, no. 6, pp. 639–653, 2007. View at MathSciNet
  23. P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser, Basel, Switzerland, 2007. View at MathSciNet
  24. A. Qian and C. Li, “Infinitely many solutions for a Robin boundary value problem,” International Journal of Differential Equations, vol. 2010, Article ID 548702, 9 pages, 2010. View at Zentralblatt MATH · View at MathSciNet
  25. J. Zhao, Structure Theory of Banach Spaces, Wuhan University Press, Wuhan, China, 1991.
  26. M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, Mass, USA, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  27. A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” Journal of Functional Analysis, vol. 14, pp. 349–381, 1973. View at Zentralblatt MATH · View at MathSciNet